The torment Daniel goes through in the first movie: he is beaten up by Johnny at a beach party, losing the friends he had just made. Then he has to go to his first day of school with a black eye, where he's banned from soccer after being tripped by one of Johnny's gang and daring to strike back. Then Johnny's gang run him off a steep hill on his bike. It culminates in a savage beating from Johnny and his friends that may well have ended up killing Daniel if Miyagi hadn't intervened.
Mr Miyagi's entire backstory, from leaving the one he loved to leave a death fight, to marrying but forced to go to war thanks to racial prejudices, only to lose his wife and unborn baby to childbirth while in combat.
Yukie explaining to Miyagi that she never married after he left Okinawa.
The death scene of Miyagi's father. The last thing he wants from both Miyagi and Sato is for them to not fight each other. And Sato, who is a massive Jerkass, can't bring himself to say no. Yet his answer is no all the same, for after the father dies, he gives Miyagi three days to mourn before Miyagi should "prepare to join him".
Miyagi trying to make peace with Sato. He asks an Armor-Piercing Question to his old friend when he ruthlessly rebuffs attempts. It's a powerful tear jerker because you can see both Miyagi's concern and Sato's pain from his long-held belief that his old friend betrayed him. It is also the first and only time Miyagi talks to anyone without his trademark broken English or referring to himself in third person.
Miyagi: What has happened to you?!
A blink-and-you'll-miss-it moment in the third film (which overlaps with Heartwarming). Miyagi takes out a picture of himself and Daniel, who has drifted away from him and has confessed that he feels 'confused'. Miyagi looks at the photo and softly utters two lines:
Miyagi: Hope confusion end soon, Daniel-san. Miyagi heart empty without you.